Consider the traffic flow diagram that follows where a1 a2 a

Consider the traffic flow diagram that follows, where a1, a2, a3, a4 are fixed positive integers. Set up linear system in the unknowns and use row operations to deduce that the system will be consistent only if

(linear algebra)

6. Consider the traffic flow diagram that follows, where ai ,a2,a3M4 b1 b2. b, b4 are fixed positive integers. Set up a linear system in the unknowns xi, r2, x3, xa and use row operations to deduce that the system will be consistent only if a1 +a2 + a3 + a4 = b1 + b2 + b3 + b4 bi X2 x4 bs b2 Additionally, find the parametric form of the solution to the system

Solution

We set up a system of linear equations as under by assuming that the traffic coming at each of the crossroads is equal to the traffic going out. Then:

a₁+ x₁ = b₁ + x_{2 ...(1)}

a₂ + x₂ = b₂ + x_{3... (2)}

a₃ + x₃ = b₃ + x_4...(3)

a₄ + x₄ = b₄ + x_1...(4)

On adding the above equations,we get ( a₁+a₂+a₃+a₄ ) + (x₁+x₂+x₃+x₄ ) = ( b₁+b₂+b₃+b₄)+(x₂+x₃+x₄+x₁ ) or, ( a₁+a₂+a₃+a₄ ) + (x₁+x₂+x₃+x₄ ) = ( b₁+b₂+b₃+b₄)+(x₁+x₂+x₃+x₄ ). Apparently, the linear equations (1) to (4) will be consistent only if ( a₁+a₂+a₃+a₄ ) =  ( b₁+b₂+b₃+b₄).

Let x₁=t.Then, from the above equations(1)to(4),we have x₄=(b₄-a₄)+t,x₃ = (b₃-a₃ )+x₄=(b₃- a₃ )+( b₄-a₄)+ t, x₂= ( b₂ - a₂ ) + x₃ = ( b₂ - a₂ ) + (b₃ - a₃ )+ ( b₄ - a₄)+ t